Structural breaks in panel data: large number of panels and short length time series Article Accepted Version Antoch, J., Hanousek, J., Horváth, L., Hušková, M. and Wang, S. (2019) Structural breaks in panel data: large number of panels and short length time series. Econometric Reviews. ISSN 1532-4168 doi: https://doi.org/10.1080/07474938.2018.1454378 Available at http://centaur.reading.ac.uk/79661/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . To link to this article DOI: http://dx.doi.org/10.1080/07474938.2018.1454378 Publisher: Taylor & Francis All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in
Transcript
Structural breaks in panel data: large number of panels and short length time series
Article
Accepted Version
Antoch, J., Hanousek, J., Horváth, L., Hušková, M. and Wang, S. (2019) Structural breaks in panel data: large number of panels and short length time series. Econometric Reviews. ISSN 1532-4168 doi: https://doi.org/10.1080/07474938.2018.1454378 Available at http://centaur.reading.ac.uk/79661/
It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing .
To link to this article DOI: http://dx.doi.org/10.1080/07474938.2018.1454378
Publisher: Taylor & Francis
All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in
Large number of panels and short length time series
Jaromır Antoch, Jan Hanousek, Lajos Horvath
Marie Huskova, Shixuan Wang
Affiliation:JA & MH, Charles University, Department of Probability and Mathematical Statistics,Sokolovska 83, CZ-18600 Praha, Czech Republic;[email protected], [email protected]
JH, CERGE-EI, a joint workplace of Charles University and the Economics Instituteof the Czech Academy of Sciences, Prague, Czech Republic; and C.E.P.R., London;[email protected]
LH, University of Utah, Department of Mathematics, Salt Lake City, UT 84112–0090USA; [email protected]
SW, Cardiff Business School, Cardiff University, Cardiff, CF10 3EU, UK; and Departmentof Economics, University of Birmingham, Birmingham, B15 2TT, UK;shixuan [email protected]
Abstract : The detection of (structural) breaks or the so called change point problem hasdrawn increasing attention from the theoretical, applied economic and financial fields.Much of the existing research concentrates on the detection of change points and asymp-totic properties of their estimators in panels when N , the number of panels, as well asT , the number of observations in each panel are large. In this paper we pursue a differ-ent approach, i.e., we consider the asymptotic properties when N → ∞ while keepingT fixed. This situation is typically related to large (firm-level) data containing finan-cial information about an immense number of firms/stocks across a limited number ofyears/quarters/months. We propose a general approach for testing for break(s) in thissetup. In particular, we obtain the asymptotic behavior of test statistics. We also pro-pose a wild bootstrap procedure that could be used to generate the critical values of thetest statistics. The theoretical approach is supplemented by numerous simulations andby an empirical illustration. We demonstrate that the testing procedure works well inthe framework of the four factors CAPM model. In particular, we estimate the breaksin the monthly returns of US mutual funds during the period January 2006 to February2010 which covers the subprime crises.
JEL classification codes : C10, C23, C33.
Key words : Change point problem; stationarity; panel data; bootstrap; four factor CAPMmodel; US mutual funds.
Acknowledgement : We thank the editor and three anonymous referees for their care-ful reading and valuable comments that helped us to improve the paper. This projectwas supported by grant GACR 15–09663S, ESRC grant ES/J50001X/1, and a RoyalEconomic Society Junior Fellowship.
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1. Introduction
Structural changes and model stability in panel data are of general concern in empiricaleconomics and finance research. Model parameters are assumed to be stable over timeif there is no reason to believe otherwise. It is well-known that various economic andpolitical events can cause structural breaks in financial data. Such events include, forexample, the change of parameters associated with the introduction of a single currencyin Europe, price liberalization in emerging markets, and world integration of equity mar-kets. Further examples and discussions can be found in Hansen (2001), Andreou andGhysels (2009) among others. The exact time of change is usually unknown and there-fore the construction of tests for stability is a challenging question. Andrews (1993) (cf.corridendum Andrews, 2003) provides the first econometric insights into break detec-tion. Brodsky and Darkhovskii (2000) and Csorgo and Horvath (1997) cover the earlydevelopment from statistical point of view.
In both the statistics and econometrics literature we can find very many of papersrelated to the detection of changes and structural breaks. Recent developments in thefield consider a general class of time series processes and panel data models with errorsthat can exhibit temporal dependence, heteroskedasticity, trending variables, possibleunit roots, cointegration and long memory processes. For a literature survey we refer thereader to Arellano (2004), Perron (2006) and Aue and Horvath (2012).
A model for panel data with possible breaks in the mean was introduced by Joseph andWolfson (1992, 1993). Bai (2010) significantly generalized the earlier results, still assum-ing that the panels are independent of each other, thanks to allowing serial correlationwithin each panel. This approach was extended by Horvath and Huskova (2012), whodiscussed dependence between the panels via common factors. Results on breaks in themeans and/or in the variances have been extended to various forms of linear regressionby Kim (2011), Baltagi et al. (2016) and Kao et al. (2012, 2014), among others.
Existing research mostly concentrates on testing for structural breaks in the panel datawhen T , the number of observations in each panel, goes to infinity and N , the number ofpanels/firms/assets/individuals, also goes to infinity, but slower than N (Bai, 2010, Baiand Carrion-i-Silvestre, 2009, Kim, 2011, Baltagi at al., 2016).
In this paper we study a different asymptotic case when T is finite and N goes toinfinity. This situation is typically related to large firm-level data panels containingfinancial information about a very substantial number of firms/stocks across a limitednumber of years/quarters. The issue of model stability in such a setup applies to broadlydefined research in empirical asset pricing as well as in corporate finance. For example,asset pricing models hinge heavily on the stability of factor parameters to explain stockreturns, while the financial industry eyes the generation of alpha returns (Fama andFrench, 1993 and 2014, Ghysels, 1998, Guercio and Reuter, 2014).
Similarly, in corporate finance the leverage literature relies on several parameters toestimate target leverage ratios, and on the stability of fixed effects to show persistencein capital structure choices (DeAngelo and Roll, 2015, Lemmon at al., 2008). Pesta andPestova (2015) use this setup for an application to insurance mathematics, while Torgov-itski (2015) focuses on multiple changes in spatial functional data. Cho and Fryzlewicz(2015) and Jirak (2015) study estimates and tests for multiple change points in highdimensional data.
There is evidence of structural breaks in the financial market that affect financialindicators, including returns and volatility, changes in exchange rate regimes, asset allo-cations, equity premium, value at risk (VaR), expected shortfall (ES), credit risk models,
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etc. Recently, researchers have started to analyze structural breaks in the panel datasetup, which reveal an increasing variety of potential applications with a strong impactin financial economics (cf., for example Antoch and Jaruskova, 2017, DeAngelo and Roll,2015, Graham et al, 2015, Whited and Wu, 2006). Hence the stability of parameterestimates in panel data, and the estimation of the time of change where T is fixed andN goes to infinity have interesting applications.
In this paper we consider the model
(1.1) yi,t = x⊤i,t(βi + δiI{t ≥ t0}) + ei,t, 1 ≤ i ≤ N and 1 ≤ t ≤ T,
where xi,t =(xi,t(1), . . . , xi,t(d)
)⊤are vectors of explanatory variables (design points that
can be random or non-random), βi ∈ Rd are unknown regression (vector) parameters inthe ith panel and ⊤ denotes the transpose of vectors and matrices. Finally, we assumethat ei,t are errors that can be cross-sectionally correlated. As previously mentioned, weconsider the detection of the unknown time of change. The regression parameters in theith panel change from βi to βi + δi at unknown time t0, usually called either a changepoint or a break point.
The paper is structured as follows. Section 2 introduces our notation, settings and liststhe main assumptions. Section 3 contains the main results (detailed proofs are providedin the technical appendix). Section 4 describes the (wild) bootstrap procedure for oneof the proposed test statistics. We use Monte Carlo simulations to illustrate the finitesample properties of the test. Section 5 contains an application to real data. Finally,Section 6 concludes the paper. Proofs are presented in Section 7.
2. Notation, settings and general assumptions
First we introduce some notation that will be used through the paper. Since we aredealing with detection and estimation of an unknown change point, we will analyze variousestimators, matrices, and parameters before and after a certain time t, 1 ≤ t ≤ T .
In particular, βi,t and βi,t denote the OLS estimators computed from the first t and
the last T − t observations of the ith panel, respectively. Hence,
We also denote by “tr” the trace of a matrix and put
Zi,t = X⊤i,tXi,t, Zi,t = X⊤
i,tXi,t, t = 1, . . . , T.
2.1. Test statistics. We propose several test statistics for the change point problem inthe panel data when T is fixed and N → ∞. We are interested in testing if there is achange in the regression parameters. We introduce two processes and their functionalscan be used as test statistics.
Under the null hypothesis there is no change in the coefficients, i.e., the null hypothesisis stated as
(2.3) H0 :N∑
i=1
‖δi‖ = 0,
where ‖ · ‖ denotes the Euclidean norm vectors and matrices.We consider two types of test procedures; the first is based on quadratic forms of least
squares estimators and the second employs sums of squares of residuals. First, let usconsider
(2.4) UN (t) =
N∑
i=1
(βi,t − βi,T )⊤Ci,t(βi,t − βi,T ), max{d, t1} ≤ t ≤ min{T − d, T − t2}.
UN(t) is a Wald-type test statistic (with weighting matrices Ci,t’s) for each t to comparethe least squares estimators computed from the first t observations with the estimatorfrom the full sample. To keep the detection procedure broad we do not restrict Ci,t to aparticular form. Instead we list a set of assumptions that Ci,t should satisfy and we willdiscuss pros and cons of particular choices of Ci,t later.
The second test procedure is based on the process
(2.5) VN(t) =
N∑
i=1
t∑
s=1
e2i,s, ei,s = yi,s − x⊤i,sβi,T , 1 ≤ i ≤ N, 1 ≤ t ≤ T,
representing CUSUM-type statistics based on partial sums of the squares of the residuals.In general, large values of UN (t) and/or VN(t) for at least some t indicate that the null
hypothesis is violated. Required critical values can be obtained either asymptotically orthrough a proper version of the bootstrap.
2.2. Assumptions. This subsection contains a set of assumptions needed to prove themain theorems describing asymptotic properties of the suggested test statistics. We startwith assumptions about the error terms and assume that they can be decomposed as
(2.6) ei,t = γ⊤i Λt + εi,t,
where Λt ∈ Rq, 1 ≤ t ≤ T is the common factor, εi,t, 1 ≤ i ≤ N, 1 ≤ t ≤ T are theinnovations and γi ∈ Rq, 1 ≤ i ≤ N , is loading in the ith panel satisfying:
Assumption 2.1. EΛt = 0 and E ‖Λt‖2 < ∞, 1 ≤ t ≤ T .
Assumption 2.2. (i) The innovations εi = (εi,1, εi,2, . . . , εi,T )⊤, i = 1, . . . , N are
independent;(ii) E εi,t = 0, E εi,tεi,s = 0 and c3 ≤ σ2
i = E ε2i,t ≤ c4 for all 1 ≤ i ≤ N, 1 ≤ t 6= s ≤ Twith some constants 0 < c3 ≤ c4 < ∞;
4
(iii) there is κ > 4 such that lim supN→∞1N
∑Ni=1E |εi,t|κ < ∞, 1 ≤ t ≤ T.
Note that Assumption 2.2 (ii) is usually satisfied in financial applications, since ARCH,GARCH and similar volatility models assume uncorrelated innovations
(cf. Francq and
Zakoian (2010)).
Assumption 2.3. {Λt, 1 ≤ t ≤ T} and {εi, 1 ≤ i ≤ N} are independent.
Let us note that serial correlation of the error disturbances in the original model (1.1)can be obtained via common factors defined in error term decomposition (2.6). The serialcorrelation observed in ei,t can then be viewed as impure correlation caused by omittingcommon factors in the main regression model (1.1).
Assumption 2.4.
limN→∞
1
N1/2
N∑
i=1
‖γi‖2 = 0.
As usual we assume that the common factors are negligible. Inference with non-negligible common factors in change-point detection has been mentioned by several au-thors in different setups when both N and T go to infinity but N is much smaller thanT (cf. Horvath and Huskova, 2012 and Baltagi et al., 2016). We discuss the behavior ofUN(t) and VN(t) when Assumption 2.4 fails in Section 3.4.
Now we continue with assumptions on the design matrix.
Assumption 2.5. There is c0 such that ‖xi,t‖ ≤ c0 for all 1 ≤ i ≤ N, 1 ≤ t ≤ T .
Assumption 2.6. (i) There are positive constants t1 and c1 such that(∑t1j=1 xi,jx
⊤i,j
)−1exists and ‖
(∑t1j=1 xi,jx
⊤i,j
)−1‖ ≤ c1 for all i, 1 ≤ i ≤ N ;
(ii) there are positive constants t2 and c2 such that(∑T
j=T−t2xi,jx
⊤i,j
)−1exists and
‖(∑T
j=T−t2xi,jx
⊤i,j
)−1‖ ≤ c2 for all i, 1 ≤ i ≤ N .
Notice that Assumption 2.6 (i) implies ‖(∑t
j=1 xi,jx⊤i,j
)−1‖ ≤ c1 for all t ≥ t1 and that
similarly to Assumption 2.6 (ii) we find that ‖(∑T
j=t+1 xi,jx⊤i,j
)−1‖ ≤ c2 for all t ≤ T − t2.
Assumption 2.6 (i) guaranties that the OLS estimators in (2.1) and (2.2) are uniquelydefined.
For the sake of simplicity we assume that the xi,t’s are non-random. We briefly discussmodifications of Assumptions 2.5 and 2.6 to allow random design vectors in Section 3.5.
Finally, we present our assumptions on weighting matrices.
Assumption 2.7. Ci,t = C⊤i,t, it is non–negative definite for all 1 ≤ i ≤ N, d ≤ t ≤ T−d,
and there exist constants c5 and t3, t4 such that ‖Ci,t‖ ≤ c5 for all 1 ≤ i ≤ N, t3 ≤ t ≤T − t4.
Instead of a particular choice of weighting matrixes, we consider a general conditionwhich would guarantee the validity of theorems and testing procedures specified later.Nevertheless, by a specific choice of Ci,t we can obtain variants/generalizations of stan-dard tests. In particular, by choosing
(2.7) Ci,t = Z−1i,T Zi,tDi,tZi,tZ
−1i,T ,
5
where Di,t are positive semidefinite matrices and βi,t and βi,t are defined by (2.1) and(2.2), we have that
U1,N(t) ≡ U1,N(t;Di,t) = UN (t;Z−1i,T Zi,tDi,tZi,tZ
−1i,T )(2.8)
=
N∑
i=1
(βi,t − βi,t)⊤Di,t(βi,t − βi,t).
Another useful choice of the weighting matrices C i,t is Ci,t = Zi,tGi,tZi,t gives
U2,N(t) = UN(t;Zi,tGi,tZi,t)(2.9)
=
N∑
i=1
(t∑
v=1
xi,v(yi,v − x⊤i,vβi,T )
)⊤
Gi,t
(t∑
v=1
xi,v(yi,v − x⊤i,vβi,T )
),
where Gi,t is a positive semidefinite weighting matrix.We note that U2,N(t) requires only the existence of Z
−1i,T . Also, U2,N(t) can be interpreted
as a weighted version of VN(t).
3. Main results
3.1. Results under H0. First we present the main results on the limit behavior of UN (t)and VN(t) under the null hypothesis.
Define
a2i,t,1 = σ2i
(t∑
s=1
x⊤i,s
(Z−1
i,t − Z−1i,T
)Ci,t
(Z−1
i,t − Z−1i,T
)xi,s +
T∑
s=t+1
x⊤i,sZ
−1i,TCi,tZ
−1i,Txi,s
)(3.1)
= σ2i
{tr((Z−1
i,t − Z−1i,T )Ci,t(Z
−1i,t − Z−1
i,T )Zi,t
)+ tr
(Z−1
i,TCi,tZ−1i,T Zi,T−t
)}
= σ2i tr(Ci,t(Z
−1i,t − Z−1
i,T
)),
A(1)N (t) =
∑
1≤i≤N
a2i,t,1,(3.2)
a2i,t,2 = σ2i
t∑
s=1
(1− x⊤
i,sZ−1i,Txi,s
)= σ2
i
(t− tr
(Zi,tZ
−1i,T
)),(3.3)
A(2)N (t) =
N∑
i=1
a2i,t,2,(3.4)
Si,t =t∑
s=1
xi,sεi,s.(3.5)
To state the asymptotic distributions of UN(t) and VN(t) we need the existence of theirasymptotic covariance functions. This is specified in Assumptions 3.1 and 3.2.
Assumption 3.1. The function
Γ(1)(t, t′) = limN→∞
1
N
N∑
i=1
[E{(
Z−1i,t Si,t − Z−1
i,TSi,T
)⊤Ci,t
(Z−1
i,t Si,t − Z−1i,TSi,T
)
×(Z−1i,t′Si,t′ − Z−1
i,TSi,T )⊤Ci,t′(Z
−1i,t′Si,t′ − Z−1
i,TSi,T )}− a2i,t,1a
2i,t′,1
]
6
exists for all t0 ≤ t, t′ ≤ T − t0, where t0 = max(d, t1, t3) and t0 = max(d, t2, t4).
Assumption 3.2. The function
Γ(2)(t, t′) = limN→∞
1
N
N∑
i=1
[t∑
s=1
t′∑
s′=1
E{(εi,s − x⊤
i,sZ−1i,TSi,T )(εi,s′ − x⊤
i,s′Z−1i,TSi,T )
}2 − a2i,t,2a2i,t′,2
]
exists for all 1 ≤ t, t′ ≤ T .
Using these assumptions, we can state two theorems on asymptotic distribution of thetest statistics UN (t) and VN(t) under the null hypothesis.
Theorem 3.1. If H0, Assumptions 2.1–2.7 and 3.1 hold, then we have that, as N → ∞,
(3.6){N−1/2
(UN(t)−A
(1)N (t)
), t0 ≤ t ≤ T − t0
}D→{ξ(1)t , t0 ≤ t ≤ T − t0
},
where ξ(1)t , t0 ≤ t ≤ T − t0, is jointly normal with zero mean, covariance Γ(1)(t, t′) and
t0, t0 are defined in Assumption 3.1.
Theorem 3.2. If H0, Assumptions 2.1 – 2.6 and 3.2 hold, then we have that, as N → ∞,
(3.7){N−1/2
(VN(t)− A
(2)N (t)
), 1 ≤ t ≤ T
}D→{ξ(2)t , 1 ≤ t ≤ T
},
where ξ(2)t , 1 ≤ t ≤ T, is jointly normal with zero mean and covariance Γ(2)(t, t′) defined
in Assumption 3.2.
Proofs of both theorems are given in Section 7.
Theorem 3.1 and Theorem 3.2 can be used to find the asymptotic distributions of thetest statistics based on UN (t) and VN(t) when H0 holds.
3.2. Results under alternatives. We briefly discuss the behavior of the proposed teststatistics when a change occurs. We focus on the statistics based on UN (t), t0 ≤ t ≤ t0,since the behavior of tests based on VN(t), 1 ≤ t ≤ T , is similar.
We obtain asymptotically consistent tests of rejecting H0 for large values of
N−1/2 maxt |UN(t) − A(1)N (t)|, if some of the changes δi’s, the sizes of the changes, are
sufficiently large. Particularly, if Assumptions 2.1 – 2.7 are satisfied and for some t0 ≤t, t1 ≤ T − t0
(3.8)1√N
N∑
i=1
δ⊤i
(Zit1(Z
−1i,max(t1,t)
−Z−1iT )Cit(Zi,max(t1,t) −Z−1
iT )Zit1
)δi → ∞
then we have that
(3.9) N−1/2 maxt
|UN(t)− A(1)N (t)| →P ∞,
which together with Theorem 3.1 ensures asymptotic consistency of the test. One canshow that (3.9) holds under other type of alternatives including multiple changes, gradualchanges, or when larger changes occur only in a fraction of panels.
To find an estimator for t0 we choose C i,t such that for all t and i
Z−1i,t C i,tZ
−1i,t = GT
7
with some positive definite matrix GT . Then under the alternative hypothesis the be-
havior of |UN(t)−A(1)N (t)| is asymptotically equivalent with
1√N
N∑
i=1
δ⊤i
(Zi,t0Z
−1i,T Zi,tGT Zi,tZ
−1i,TZi,t0
)δi, t ≥ t0,
1√N
N∑
i=1
δ⊤i
(Zi,t0Z
−1i,TZi,tGTZi,tZ
−1i,T Zi,t0
)δi, t ≤ t0.
Note that the elements of the sums defined above are non-decreasing for t ≤ t0, whilethey are non-increasing for t ≥ t0. Then the condition for the asymptotic consistency(3.8) can be replaced by
1√N
N∑
i=1
δ⊤i
(Zit0Z
−1i,T Zit0GT Zi,t0Z
−1i,TZit0
)δi → ∞,
and an estimator tN of t0 can be defined as
tN = argmaxt=d,...,T−d
∣∣UN (t)−A(1)N (t)
∣∣.
The asymptotic properties of tN , as N → ∞, are the subject of future research.
3.3. Remarks. The test procedures studied in the previous section depend on someunknown quantities. We briefly discuss their possible estimators.
The normalizing sequences A(1)N (t) and A
(2)N (t) in Theorem 3.1 and 3.2 depend on the
unknown variances σ2i , i = 1, . . . , N. To estimate them we can employ the following
options:
(1) σ2i is estimated by the sample variance of the ith panel
(3.10) σ2i =
1
T − d
T∑
t=1
(yi,t − x⊤
i,tβi,T
)2;
(2) σ2i is estimated by
(3.11) σ2
i = mind<v<T−d
1
T − 2d
{ v∑
t=1
(yi,t − x⊤
i,tβi,v
)2+
T∑
t=v+1
(yi,t − x⊤
i,tβi,v
)2}.
The estimator σ2
i is asymptotically consistent under the one change point alternative.
However, it is not stable since it includes βi,v and βi,v when they are computed from afew observations. Hence in the simulations and the data example we use σ2
i .The covariances Γ(1) and Γ(2) in Assumptions 3.1 and 3.2 are complicated functions of
the observations. Nevertheless, they can be simplified for certain choices of the weightingmatrices Ci,t and moderate values of T . It follows from Assumptions 2.2 and 2.5 that
(3.12)1
N
N∑
i=1
E
∥∥∥∥∥
t∑
v=s
xi,vεi,v
∥∥∥∥∥
4
≤ c2,1(t− s+ 1)2
8
and ∥∥∥∥∥
t∑
v=s
xi,vx⊤i,v
∥∥∥∥∥ ≤ c2,2(t− s+ 1)
where c2,1 and c2,2 are some positive numbers. We note that if the xi,t’s realizations ofstationary, ergodic sequences with finite mean, then
1
TZi,T ≈ Zi,
assuming that Zi is nonsingular, we arrive at the natural conclusion that
(3.13) ‖Z−1i,T‖ ≤ c2,3/T.
Let us assume that Ci,t = Z2i,t in the definition of UN (t). In this case
Γ(1)(t, t′) = limN→∞
1
N
N∑
i=1
E
{(Si,t − Zi,tZ
−1i,TSi,T )
⊤(Si,t − Zi,tZ−1i,TSi,T )
× (Si,t′ − Zi,t′Z−1i,TSi,T )
⊤(Si,t′ − Zi,t′Z−1i,TSi,T )− a2i,t,1a
2i,t′,1
},
and therefore by (3.12) and (3.13) we have for moderate and large T ’s that
Γ(1)(t, t′) ≈ limN→∞
1
N
N∑
i=1
E
{(Si,t −
t
TSi,T
)⊤(Si,t −
t
TSi,T
)
×(Si,t′ −
t
TSi,T
)⊤(Si,t′ −
t′
TSi,T
)− a2i,t,1a
2i,t′,1
}.
The approximation for Γ(1)(t, t′) also shows that UN(t) is related to CUSUM processesused extensively in change point analysis.
Under Assumption 2.4 (negligible influence of common factors) the residual ei,t =
yi,t − xi,tβi,T is a good approximation for the innovation εi,t. Hence, under H0, we coulduse
Γ(1)N (t, t′) =
1
N
N∑
i=1
{S⊤i,tSi,tS
⊤i,t′Si,t′ − a2i,t,1a
2i,t′,1
},
where
Si,t =
t∑
v=1
xi,vei,v, a2i,t,1 = σ2i,T tr
(Zi,tZ
−1i,T Zi,t
)and σ2
i,T =1
T
T∑
v=1
e2i,v,
to approximate the covariance matrix Γ(1)(t, t′), since Si,T = 0.
In some models commonly used in econometric finance, e.g., CAPM type models, xi,t’sdo not depend on i. The model and also the test statistics become much simpler, becausethe indices i in xit’s and Zit’s can be omitted and our weighting matrix also becomesindependent on i, i.e., C it = Ct, i = 1, . . . , N . Hence we have
A(1)N (t) = tr(Ct(Z
−1t −Z−1
T ))
N∑
i=1
σ2i and A
(2)N (t) = (t− tr(ZtZ
−1T ))
N∑
i=1
σ2i .
9
Then it suffices to estimate∑N
i=1 σ2i instead of a set of single σ2
i ’s, i = 1, . . . , N .
3.4. Non-negligible common factors. In this section we show that if Assumption 2.4
(negligible common factors) fails, then N−1/2|UN(t) − A(1)N (t)| → ∞ and N−1/2|VN(t) −
A(2)N (t)| → ∞ in probability for some t. These are immediate consequences of Theorem 3.3
and 3.4 stated below.
Assumption 3.3.
1
rN
N∑
i=1
‖γi‖2 = O(1) with some numerical sequence satisfying rN/N1/2 → ∞.
Assumption 3.4. The limit
Q(s, v, z) = limN→∞
1
rN
N∑
i=1
γix⊤i,s
(Z−1
i,z − Z−1i,T
)Ci,z
(Z−1
i,z − Z−1i,T
)xi,vγ
⊤i , 1 ≤ s, v ≤ z ≤ T,
exists and there are s0, v0 and z0 such that 1 ≤ s0, v0 ≤ z0 ≤ T , and at least one elementof Q(s0, v0, z0) is different from 0.
Assumption 3.5. The limit
Q(s, v, z) = limN→∞
1
rN
N∑
i=1
(1
Tγi − γix
⊤i,sZ
−1i,Txi,z
)(1
Tγi − γix
⊤i,vZ
−1i,Txi,z
)⊤
, 1 ≤ s, v ≤ z ≤ T,
exists, and there are s0, v0 and z0 such that 1 ≤ s0, v0 ≤ z0 ≤ T , and at least one elementof Q(s0, v0, z0) is different from 0.
It follows from Assumption 3.3 that ‖Q(s, v, z)‖ < ∞. Similarly, Assumption 3.5implies that ‖Q(s, v, z)‖ < ∞. The rate rN is the exact rate in Assumptions 3.3 and 3.4,i.e. the matrix Q is different from the zero matrix. The term describing the dependencebetween the panels dominates the limit since rN/N
1/2 → ∞.
Theorem 3.3. If H0 and Assumptions 2.1–2.3, 2.5 – 2.7, 3.1 and 3.4 hold, then we have(3.14){
1
rN
(UN (t)− A
(1)N (t)
), t0 ≤ t ≤ T − t0
}D→{ξ(3)t , t0 ≤ t ≤ T − t0
}, N → ∞,
where
ξ(3)t =
t∑
s,v=1
Λ⊤s Q(s, v, t)Λv, t0 ≤ t ≤ T − t0,
non-zero matrix Q is defined in Assumption (3.4) and t0, t0 are defined in Assumption 3.1and rN in Assumption (3.3).
Now we formulate the analogue of Theorem 3.3 for VN(t).
Theorem 3.4. If H0 and Assumptions 2.1–2.3, 2.5 – 2.6 and 3.5 hold, then we have
(3.15)
{1
rN
(VN(t)− A
(2)N (t)
), 1 ≤ t ≤ T
}D→{ξ(4)t , 1 ≤ t ≤ T
}, N → ∞,
where
ξ(4)t =
t∑
s,v=1
Λ⊤s Q(s, v, t)Λv, 1 ≤ t ≤ T,
10
and the non-zero matrix Q is defined in Assumption 3.5.
Proofs of both theorems are given in Section 7.
Under the Assumptions of Theorems 3.3 and 3.4 the limit distributions are fully de-termined by the common factor Λs, s ∈ T, and are not normally distributed.
3.5. Random design vectors. For the sake of simplicity, Theorem 3.1 and 3.2 withtheir proofs are provided for non-random xi,t’s. However, the results can be extended torandom design points. We assume:
Assumption 3.6. {εi,t,Λt, 1 ≤ i ≤ N, 1 ≤ t ≤ T} and {xi,t, 1 ≤ i ≤ N, 1 ≤ t ≤ T}are independent.
Assumptions 2.5 and 2.6 are modified so the uniform boundness is replaced with theuniform boundness of moments:
Assumption 3.7. There is c0 such that E||xi,t||8 ≤ c0 for all 1 ≤ i ≤ N, 1 ≤ t ≤ T
and
Assumption 3.8. (i) There are positive constants c1 and t1 such that E||(∑t1
t=1 xi,txTi,t)
−1||8 ≤c1 for all 1 ≤ i ≤ N .
(ii) There are positive constants c2 and t2 such that E||(∑Tt=T−t2
xi,txTi,t)
−1||8 ≤ c2 forall 1 ≤ i ≤ N .
It follows from Assumption 3.8 that
P
(( t∑
t=1
xi,txTi,t
)−1
exists for all t1 ≤ t ≤ T
)= 1
and
P
(( T∑
t=T−t
xi,txTi,t
)−1
exists for all t2 ≥ t
)= 1.
We also need to replace a2i,t,1 and a2i,t,2, 1 ≤ i ≤ N, to ≤ t ≤ T − t0 with their expectedvalues.
4. Bootstrap tests and simulations
4.1. Bootstrap tests. In this subsection we describe the settings of the wild bootstrapprocedure that could be used to test for the change point.
Due to Assumption 2.4 (negligible common factors) we could also apply the bootstrapto get approximations to the distribution functions of UN(t) and VN(t). We demonstratethe application of the wild bootstrap for UN (t), and VN(t) can be resampled in a similarway.
For t0 ≤ t ≤ T − t0 and 1 ≤ i ≤ N let
φi,t = (βi,t − βi,T )⊤Ci,t(βi,t − βi,T )−
1
N
N∑
j=1
(βj,t − βj,T )⊤Cj,t(βj,t − βj,T ).
Assumption 4.1. ζ1, . . . , ζN are independent standard normal variables, independent of{εi,t,Λt, 1 ≤ i ≤ N, 1 ≤ t ≤ T}
11
The bootstrapped observations are defined by
(4.1) φ∗i,t = ζi φi,t, t0 ≤ t ≤ T − t0, 1 ≤ i ≤ N.
and the respective bootstrapped process by
(4.2) U∗N (t) = N−1/2
N∑
i=1
φ∗i,t, t0 ≤ t ≤ T − t0.
Theorem 4.1. We assume that Assumptions 2.6 – 3.1 and 4.1 hold.(i) If H0 holds, then
{U∗N(t), t0 ≤ t ≤ T − t0} D→ {ξ(1)t , t0 ≤ t ≤ T − t0}
where ξ(1)t , t0, t0 are defined in Theorem 3.1.
(ii) If HA holds, thensup
t0≤t≤T−t0
|U∗N(t)| = OP (1).
Proof: See Section 7.
4.2. Test procedure and algorithm. Based on the results in the previous sections wedescribe here the test procedures for testing H0 specified in (2.3) of our model (1.1). Ourtest procedures are based on the maximum functionals of
(4.3) UN(t) = N−1/2(UN(t)− A(1)N (t))
where UN(t) and A(1)N (t) is defined in (2.4) and (3.2), respectively, and
(4.4) UN (t) = N−1/2(UN(t)− A(1)N (t))
with
A(1)N (t) =
N∑
i=1
a2i,t,1, a2i,t,1 = σ2i tr(C i,t(Z
−1i,t −Z−1
i,T ))
and for σ2i we recommend the estimator defined in (3.10). In our study we use the test
statisticsuN = max
t|UN(t)|, and uN = max
t|UN (t)|.
Large values of either test statistics indicate that the null hypothesis is violated. Therespective critical values are constructed via wild bootstrap. Notice that uN depends onσ2i ’s when these are replaced by their estimators σ2
i ’s. Since σ2i ’s are usually unknown
the test statistic uN is more practical. In application we use uN , but in the simulationswe also present selected results for uN in order to see influence of unknown σ2
i ’s.Bellow we provide the algorithm for the test based on uN and level α
Algorithm
(1) Compute A(1)N (t), UN (t) and uN as described above and σ2
i using formula (3.10).(2) Generate bootstrap version u⋆
N :(a) Calculate φi,t using
φi,t = (βi,t − βi,T )⊤Ci,t(βi,t − βi,T )−
1
N
N∑
j=1
(βj,t − βj,T )⊤Cj,t(βj,t − βj,T ).
12
(b) Generate ζi from standard normal distribution N(0, 1).(c) Generate bootstrap observations φ∗
i,t using relation
φ∗i,t = ζi φi,t, t0 ≤ t ≤ T − t0, 1 ≤ i ≤ N.
(d) Obtain bootstrap process
U∗N (t) =
1√N
N∑
i=1
φ∗i,t.
(e) Get the bootstrap version u⋆N by taking maximum values of U∗
N(t), i.e.
u⋆N = max
d<t<T−d
∣∣U∗N(t)
∣∣.
(3) Find critical values from bootstrap.• Repeat step (2) B = 1000 times, obtain (1 − α)100% quantile value of u⋆
N
and denote it u⋆N,α.
• Reject H0 if uN > u⋆N,α.
4.3. Simulations. For our simulations we used the model
yi,t = x⊤i,t
(βi + δiI{t ≥ t0}
)+ ei,t, 1 ≤ i ≤ N, 1 ≤ t ≤ T.
with x⊤i,t =
(1, xi,t(2), . . . , xi,t(d)
)⊤, d = 2 and 5. The xi,t(j)’s, j = 2, . . . , d, has normal
distribution N(1, 1), and are independent of ei,t, 1 ≤ t ≤ T, 1 ≤ i ≤ N .The errors ei,t, t = 1, . . . , T, are independent sequences. We consider three cases:
• The “iid case”: ei,t, i = 1, . . . , N, t = 1, . . . , T, are independent and identicallydistributed and follow standard normal distribution.
• The “unequal variances”: ei,t, i = 1, . . . , N, t = 1, . . . , T, are independent andnormally distributed with zero mean and variance σ2
i , that are generated fromuniform distribution U(0.5, 1.5).
• The “GARCH(1,1) process”: ei,t = νi,tεi,t, where ν2i,t = 0.25+0.25e2i,t−1+0.5ν2
i,t−1
and εi,t are independent and follow the standard normal distribution N(0, 1).
In the sequel we report on two model specifications:
S1: Linear model with d = 2, β = (1, 2)⊤ and Ci,t = ZitZit.S2: Linear model with d = 5, β = (1, 1, 2, 3, 4)⊤ and Ci,t = Zi,tZ
−1i,TZi,t.
We use extensive simulations to explore finite sample properties of the underlying testprocedure. In particular, we analyze the significance level and the empirical power of thetest for different pairs of (N, T ), different assumptions on t0 (the change point) and variousassumptions on the errors terms. Overall, we keep the number of simulations equal to1 000 and the number of bootstrap steps also equals 1 000. In our simulation setup wealso include cases when both N and T are small, to explore small sample properties ofthe test statistics uN and uN and the behavior of the proposed wild bootstrap procedure.In simulations we first consider the case when non zero changes occur in all panels at thesame time t0. Later we analyze the sensitivity of the detection of change in the case whenonly a fraction of panels (ϑ = 0.25) experience change. Selected results are presentedbellow.
A. Empirical versus theoretical significance level
Our results show that, even for rather small N and T the nominal rejection rate was closeto the corresponding significance level of the tests. Table 4.1 demonstrates how well the
13
nominal rejection of the null hypothesis (no change) in the case of model S2 corresponds tothe chosen significance level, with four independent explanatory variables and intercept.A similar result is reported in Table 4.2 for model S1 for small N and T . Other modelassumptions (different number of regressors, different data generating process, differentdistributions) lead to similar results; additional results are available upon request.
Table 4.1. Nominal rejection of the H0; test based on uN (left part of thetable) and on uN (right part of the table) at significance level 5%; model S2(four regressors and intercept).
σ2i known (uN) σ2
i estimated (uN)
number of panels number of panels100 200 500 1000 100 200 500 1000
Table 4.2. Nominal rejection of the H0; test based on uN (left part of the table) andon uN (right part of the table) at significance level 5%; model S1 (one regressor andintercept).
B. Empirical power of the test statistic uN
In our simulation setup we set the time of change t0 equal to T/2 and T/4 under HA.Table 4.3 shows the empirical power of the tests based on uN and uN at a 5% significancelevel in case of model S2, for several combinations of N and T . The empirical power isan increasing function of N and T . It is high even for small and moderate N and T ,which can be explained by changes in all panels at the same time. We note that theestimation of the σ2
i ’s reduces power but very mildly. By increasing N for each T we seethe increasing empirical power of the test. Similar results of the simpler model S1 withdifferent data generating processes are depicted in Table 4.4
Table 4.3. Power of the test based on uN when the change occurs at t0 = T/2or T/4 in all regressors of model S2 (δ = (0, 1, 0.3, 0.2, 0.1)⊤); the significancelevel 5%.
change in intercept change in slope
number of panels number of panels100 200 500 1000 100 200 500 1000
Table 4.4. Power of the test based on uN when the change occurs at t0 = T/2in either intercept or slope of model S1 (δ = 0.1); the significance level 5%.
C. Sensitivity analysis to fraction of panels experiencing a change
We also investigate a situation when only a fraction of the panels experience a change.The fraction of the panels with change is denoted by ϑ. Given the space limitations wepresent this sensitivity analysis only for the model S1 and for the case of uN
(σ2i estimated
by σ2i
). The other setups lead to similar results. Bellow we present the results for model
S1 when all changes occurred at time t0 = T/2. We applied uN to detect instability at a5% significance level and the size of changes was δ = (0.25, 0) (change in intercept) andδ = (0, 0.25) (change in the slope). The results are presented in Table 4.5 when ϑ = 0.25.For other ϑ and δ not presented here we observed the following pattern. The power of thetest decreases with ϑ (fraction of panels with changes) and increases with δ (size of thechange). The level of detection is very good, even if only a quarter of the panels containschanges in the regression parameters; one can see that the power of the test increaseswith T and with N . Additional results are available upon request.
15
change in intercept change in slope
number of panels number of panels100 200 500 1000 100 200 500 1000
Table 4.5. Power of the test based on uN when the change occurs at t0 = T/2in either intercept or slope of model S1 (δ = 0.25); fraction of panels withchange ϑ = 0.25; the significance level 5%.
5. Empirical Illustration
In this section we present an application of our theoretical results to the capital assetpricing model (CAPM). We apply our approach to the CAPM, using the conventionalfactor model, which includes the Fama-French three-factor model augmented with theCarhart (1997) momentum factor.
Specifically, we test for breaks in coefficients for the US mutual fund return data aroundthe sub-prime crisis. Our time span covers the period from January 2006 to February2010, which is consistent with other sub-prime crisis studies (e.g. Dick-Nielsen et al.,2012). The sub-prime crisis is generally defined as the period between the fourth quarterof 2007 and the end of 2008 (Santos, 2011).
The Fama-French (1993) three factor model augmented with the Carhart (1997) mo-mentum factor is defined as
Ri,t − Rft = αi,t +
(RM
t −Rft
)βMi,t +RHML
t βHMLi,t +RSMB
t βSMBi,t +RMOM
t βMOMi,t + ei,t,
1 ≤ t ≤ T, 1 ≤ i ≤ N , where Ri,t − Rft denotes the excess return on the mutual fund;
RMt −Rf
t is the market risk premium; RHMLt is the value factor, calculated as the return
difference between portfolios with the highest decile of stocks and the lowest decile ofstocks in terms of the ratio of book equity-to-market equity (HML); RSMB
t is the valuefactor, calculated as the return difference between portfolios with the smallest decile ofstocks and the largest decile of stocks in terms of size (SMB); RMOM
t is the momentumfactor calculated as the return difference between portfolios with the highest decile ofstocks and lowest decile of stocks in terms of recent return (i.e., momentum, or MOM );and ei,t the random errors.
In our investigation we use uN as our test statistics and we display the graphs of
|UN(t)|. The unknown σ2i is estimated by σ2
i of (3.10). The critical values were obtainedby the wild bootstrap of Section 4.1.
16
5.1. Data Information. The four factors can be downloaded from Ken French’s datalibrary1. The raw dataset on mutual funds contains monthly return data of 6 190 USmutual funds from January 1984 to November 20142. However, there are many missingvalues in the mutual fund dataset because different mutual funds have different start datesand some of them have already been terminated. For the purpose of our illustration weselect the mutual funds that have no missing returns for the period of the subprime crisis(January 2006 to February 2010).
Using the Yahoo finance classification, we consider nine categories of the US mutualfunds. These are Large Blend, Large Growth, Large Value, Middle Blend, Middle Growth,Middle Value, Small Blend, Small Growth and Small Value. These categories are combi-nations of the mutual fund size and their investment strategies as illustrated in Table 5.1.
LargeMediumSmall
Blend Growth Value
Table 5.1. Categories of mutual funds.
5.2. Results for the sub-prime crisis. Table 5.2 shows the test results for the sub-prime crisis. It covers the period from January 2006 to February 2010, which is equal to50 periods. uN is the test statistic, CV stands for critical values which are reported for10%, 5% , and 1% significance levels obtained by the wild bootstrap. The break pointcolumn shows the month (time period) in which a break point was detected and N showsthe number of observations per fund category. Between the brackets in the breakpointcolumn the estimated time of the change is reported.
For all but one fund categories the test statistic uN is above the 5% critical valuesduring the sub-prime crisis period (middle of 2008 to early 2009). Interestingly, the testcannot detect change for the category of Small Value mutual funds at the 5% significancelevel. However the Small Value mutual fund category is significant at the 10% significancelevel.
The coefficients for the four factors changed first for the Large Blend mutual fundcategory with estimated break point in August 2008, and the Small Growth mutualfund category has estimated change point just a month later in September 2008. Thecoefficients for Large Growth, Middle Blend, Middle Growth, and Small Blend changedlast, in March 2009.
It is interesting to note that only the Large Blend indicated a change in the environmentout of all large-type mutual funds. Recall that the Large Blend is defined as a balancedmix of growth and value stocks. We hypothesize that the Large Blend more closelyresembles the market and that is why it indicated a shift in the underlying structure.
1The factors data is available athttp://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html
2The monthly return history of US mutual funds is available for the period January 1984 to November2014 at the web site http://finance.yahoo.com
17
Category uN 10% CV 5% CV 1% CV break point t0 N
Large Blend 47 125 18 361 21 964 28 961 Aug 2008 (32) 659
Large Growth 149 624 36 993 43 449 56 892 Mar 2009 (39) 748
Large Value 48 210 9 138 10 576 13 239 Feb 2009 (38) 528
Table 5.2. Detection of the break point during the sub-prime crisis
Figure 5.1 plots the estimated break points for the coefficients for the Mutual Fundcategories with the levels of the S&P 500. The model employed is the Fama and Frenchthree factor model (1993) augmented with the Carhart (1997) momentum factor.
0
200
400
600
800
1000
1200
1400
1600
1.0
8
2.0
8
3.0
8
4.0
8
5.0
8
6.0
8
7.0
8
8.0
8
9.0
8
10
.08
11
.08
12
.08
1.0
9
2.0
9
3.0
9
4.0
9
5.0
9
6.0
9
7.0
9
8.0
9
9.0
9
10
.09
11
.09
12
.09
1.1
0
S&
P 5
00
S&P 500 and Break Points for Mutual Funds
Large Blend
Small Growth
Middle Value
Large Value
Large Growth
Middle Blend
Middle Growth
Small Blend
Figure 5.1. Estimated break points of the mutual fund returns and the level of theS&P 500.
The Large Blend category indicated a structural change when the S&P 500 was wellabove the 1200 value. This could have been used as a potential trading signal. The lowestpoint of the S&P 500 was February 2009, below the 800 mark. Here the change point wasdetected for the Large Growth, Middle Blend, Middle Growth, and Small Blend. Again,as for the earlier estimated break point for Large Blend, this time could also be used asa trading signal, obviously indicating an upturn of the market.
Figures 5.2 – 5.10 graphically depict the processes∣∣ UN(t)
∣∣, estimated break point tN ,and the estimated simulated critical values at the 10%, 5% , and 1% significance levelsfor all categories of mutual funds.
18
2006 2007 2008 2009 20100
2
4
6104
1% CV5% CV10% CV
Figure 5.2. Graphs of∣∣ UN (t)
∣∣ and simulated critical values for Large Blend mutualfunds.
Figure 5.2 shows∣∣ UN(t)
∣∣ and simulated critical values for the category of Large Blend
mutual funds.∣∣ UN (t)
∣∣ stays below the critical level up until late 2007 and then in 2008 it
starts slowly breaking off. The estimated break point(corresponding to the highest value
of∣∣ UN (t)
∣∣) is in August of 2008, but it is clear that in the preceding months an increase
is visible. After the break in the coefficients,∣∣ UN (t)
∣∣ slowly decreased to regular levelsin late 2009 and early 2010. Another peak in the first quarter of 2009, could suggest aninfluence of the European sovereign debt crisis, i.e., potentially two break points.
2006 2007 2008 2009 20100
5
10
15104
1% CV5% CV10% CV
Figure 5.3. Graphs of∣∣ UN (t)
∣∣ and simulated critical values for Large Growth mutualfunds.
Figure 5.3 shows∣∣ UN(t)
∣∣ and simulated critical values for the category of Large Growthmutual funds. The estimated break point is in March of 2009, which is at the deepest
point of the sub-prime crisis. However, the graph tells a slightly different story.∣∣ UN (t)
∣∣is at zero in 2006 and 2007, and slowly increases in 2008. The process
∣∣ UN (t)∣∣ is well
above the simulated critical values in late 2008 and all 2009, but the estimator of thetime of a change is quite late due the magnitude of the March 2009 value. Immediatelyafter the detection,
∣∣ and simulated critical values for Large Value mutualfunds.
Figure 5.4 shows∣∣ UN (t)
∣∣ and simulated critical values for the category of Large Valuemutual funds. The estimated break point is in February 2009. Obviously large mutualfunds could be strongly influenced by the European sovereign debt crisis that startedin early 2009. Again, a similar story emerges to the Large Value mutual fund. Values
of∣∣ UN (t)
∣∣ are around zero and well below the simulated critical values in 2006, 2007,
and early 2008. For most of 2008 the process∣∣ UN(t)
∣∣ is above the simulated criticalvalues but the break point is again indicated late. After the estimated break in February∣∣ UN(t)
∣∣ drops below critical value levels.
2006 2007 2008 2009 20100
5
10104
1% CV5% CV10% CV
Figure 5.5. Graphs of∣∣ UN(t)
∣∣ and simulated critical values for Middle Blend mutualfunds.
2006 2007 2008 2009 20100
0.5
1
1.5
2105
1% CV5% CV10% CV
Figure 5.6. Graphs of∣∣ UN(t)
∣∣ and simulated critical values for Middle Growthmutual funds.
20
2006 2007 2008 2009 20100
1
2
3
4104
1% CV5% CV10% CV
Figure 5.7. Graphs of∣∣ UN(t)
∣∣ and simulated critical values for Middle Value mutualfunds.
Figures 5.5 – 5.7 show∣∣ UN(t)
∣∣ and simulated critical values for the category of MiddleBlend, Middle Growth, and Middle Value mutual funds. The Middle Blend and MiddleGrowth mutual fund categories have an estimated break point in March 2009. For both
mutual fund categories,∣∣ UN (t)
∣∣ remains fairly low up until early 2008. Even though thetest statistic was above the simulated critical values in late 2008 and early 2009 the break
point is indicated fairly late, in March 2009 for both funds.∣∣ UN (t)
∣∣ decreases to belowsimulated critical values shortly after the break detection. Middle Blend and MiddleGrowth mutual funds show a very similar pattern, suggesting two breakpoints. The firstrelates to to the US crisis and the second to the European sovereign debt crisis. TheEuropean crisis is detected as the main break point for Middle Blend and Middle Growth
mutual funds. Similar characteristics of∣∣ UN (t)
∣∣ are observed. The values remain verylow and well below the simulated critical values up until late 2008. After the estimated
break the values of∣∣ UN(t)
∣∣ remain quite high for a few months before dropping in late2009.
2006 2007 2008 2009 20100
1
2
3
4104
1% CV5% CV10% CV
Figure 5.8. Graphs of∣∣ UN (t)
∣∣ and simulated critical values for Small Blend mutualfunds.
21
2006 2007 2008 2009 20100
2
4
6
8104
1% CV5% CV10% CV
Figure 5.9. Graphs of∣∣ UN (t)
∣∣ and simulated critical values for Small Growth mutualfunds.
Figures 5.8 – 5.10 show∣∣ UN (t)
∣∣ and simulated critical values for the category of SmallBlend, Small Growth, and Small Value mutual funds. The estimated break points for theSmall Blend and Small Growth mutual funds are detected in March 2009 and September2008, respectively. As is clearly visible, both graphs suggest possibility of two breakpoints, as before. One is linked to the US mortgage crisis and the other to the Europeansovereign debt crisis. The test statistics for the maximum at these peaks are very similar.The simulated critical values were low until late 2008 and well below the significancelevels.
2006 2007 2008 2009 20100
2
4
6104
1% CV5% CV10% CV
Figure 5.10. Graphs of∣∣ UN(t)
∣∣ and simulated critical values for Small Value mutualfunds.
The Small Value mutual fund category does not indicate a change point because thetest statistic never exceeded the 5% critical value. However, a change is detected at the10% level.
Overall, the result shows the sensitivity of large and medium mutual funds to the 2008economic crisis, as well as the subsequent European sovereign debt crisis. For a detailedfinancial analysis of the associated change point it would be useful to classify the mutualfunds according to their international portfolio holdings. The portfolio structure and itsdiversification could likely be attributed to a possible change. We leave these points forfurther research.
6. Conclusion and suggestions for further research
The detection of (structural) breaks has been at the center of both theoretical andapplied economic and financial research for many years. Part of the existing researchconcentrates on the detection of breaks in panels when both N , the number of panels
22
and T , the time dimension are large. In contrast, in this paper we concentrate on a qual-itatively different situation. We elaborate a general approach for testing for the break(s)when N is large and T is fixed and rather small. This situation is typically relatedto large (firm-level) data containing financial information about an immense number offirms/stocks across a limited period of time. More precisely, we study the asymptoticproperties of suggested test procedures when N → ∞ while keeping T fixed. Our ap-proach also allows the estimation of the time of the change(s).
The goal of this paper is to present a background for constructing a set of quite generaltest statistics which are suitable in the given context, and to prove theorems about theirasymptotic behavior. We also aim to show that several special choices of the weightingmatrices used in our test statistics resulted in a generalization of some well-known teststatistics.
In the rich simulation section we study the finite sample properties of two processes.The maximum norm of these processes can be used as test statistics and critical valuesobtained by a wild bootstrap. We present a selected set of tables to document theaccuracy of the significance levels and the power of the test under several simulationschemes and alternatives, including cases when only a fraction of the panels experienceda break.
For a practical application we apply the testing procedure in the framework of the fourfactors CAPM model. In particular, we estimate breaks in monthly returns of US mutualfunds during the period January 2006 to February 2010 which covers the subprime crises.
The theoretical considerations concerning multiple breaks are beyond the scope of thispaper and we plan to address them in future research.
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2010.
7. Proofs
We assume that H0 holds. Using (1.1) we get
βi,t = βi + Z−1i,t
(Si,t +Ri,t
),
24
where Si,t,Ri,t ∈ Rd, Si,t is defined in (3.5) and Ri,t =∑t
s=1 xi,sγ⊤i Λs, 1 ≤ i ≤ N, 1 ≤
t ≤ T. It also holds that
(βi,t−βi,T )⊤Ci,t(βi,t − βi,T )(7.1)
= S⊤i,tCi,t,1Si,t + S⊤
i,TCi,t,2Si,T − 2S⊤i,tCi,t,3Si,T + 2S⊤
i,tCi,t,1Ri,t
+ 2S⊤i,TCi,t,2Ri,T − 2S⊤
i,tCi,t,3Ri,T − 2R⊤i,tCi,t,3Si,T +R⊤
i,tCi,t,1Ri,t
+R⊤i,TCi,t,2Ri,T − 2R⊤
i,tCi,t,3Ri,T ,
where
Ci,t,1 = Z−1i,t Ci,tZ
−1i,t , Ci,t,2 = Z−1
i,TCi,tZ−1i,T and Ci,t,3 = Z−1
i,t Ci,tZ−1i,T .
Lemma 7.1. If H0 and Assumptions 2.1 – 2.3 and 2.5 – 2.7 hold, then we have that
E S⊤i,tCi,t,1Si,t = σ2
i
t∑
s=1
x⊤i,sCi,t,1xi,s = σ2
i tr(Ci,t,1Zi,t
),
E S⊤i,TCi,t,2Si,T = σ2
i
T∑
s=1
x⊤i,sCi,t,2xi,s = σ2
i tr(Ci,t,2Zi,T
)
and
E S⊤i,tCi,t,3Si,T = σ2
i
t∑
s=1
x⊤i,sCi,t,3xi,s = σ2
i tr(Ci,t,3Zi,t
).
Proof. With the notation Ci,t,1 = {ci,t,1(ℓ, ℓ′), 1 ≤ ℓ, ℓ′ ≤ d} and Si,t = (Si,t(1), . . . , Sdi,t)
⊤
we have
S⊤i,tCi,t,1Si,t =
t∑
s=1
x⊤i,s εi,sZ
−1i,t CitZ
−1i,t
t∑
v=1
xi,vεi,v(7.2)
=
d∑
ℓ=1
d∑
ℓ′=1
Si,t(ℓ)ci,t,1(ℓ, ℓ′)Si,t(ℓ
′)
=
d∑
ℓ=1
d∑
ℓ′=1
t∑
s=1
t∑
s′=1
xi,s(ℓ)xi,s′(ℓ′)εi,sεi,s′ci,t,1(ℓ, ℓ
′),
which implies the first statement of the lemma since E εi,sεi,s′ = σ2i if s = s′, and zero
otherwise. The same argument can be used to prove the rest of Lemma 7.1.�
LetHi,t = S⊤
i,tCi,t,1Si,t + S⊤i,TCi,t,2Si,T − 2S⊤
i,tCi,t,3Si,T .
Lemma 7.2. If H0 and Assumptions 2.1 – 2.7 and 3.1 hold, then we have
{N−1/2
N∑
i=1
(Hi,t − a2i,t,1), d ≤ t ≤ T}
D→{ξ(1)t , t0 ≤ t ≤ T − t0
},(7.3)
where ξ(1)t , t0 ≤ t ≤ T − t0, is jointly normal with zero mean and covariance Γ(1)(t, t′)
and t0, t0 are defined in Assumption 3.1.
25
Proof. It follows from Assumptions 2.2, 2.6 and 2.7 that the random vectors Hi,t, t0 ≤ t ≤T − t0 are independent and by Lemma 7.1 EHi,t = a2i,t,1. Also, that E |Hi,t − a2i,t,1|κ/2 ≤cE |εi,0|κ and therefore we have
(7.4)
N∑
i=1
E |Hi,t − a2i,t,1|κ/2 = O(N).
It is easy to see that
cov(Hi,t, Hi,t′
)= E
[(Z−1
i,t Si,t − Z−1i,TSi,T
)⊤Ci,t
(Z−1
i,t Si,t − Z−1i,TSi,T
)
×(Z−1
i,t′Si,t′ − Z−1i,TSi,T
)⊤Ci,t′(Z
−1i,t′Si,t′ − Z−1
i,TSi,T )]−a2i,t,1a
2i,t′,1.
We can assume that {Γ(1)(t, t′), t0 ≤ t, t′ ≤ T − t0} is a non–singular matrix. IfΓ(1)(t, t′) is singular, one needs to prove the central limit theorem for a subvector of(Hi,t
0, . . . , Hi,T−t0
)⊤such that the related submatrix of Γ(1)(t, t′) is regular and has the
same rank as Γ(1)(t, t′). Hence for any constants(ct
0, ct
0+1, . . . , cT−t0
)6= 0 we have
limN→∞
1
N
N∑
i=1
var
T−t0∑
v=t0
cvHi,v
> 0.
Equation (7.4) implies that
N∑
i=1
E
∣∣∣∣∣∣
T−t0∑
v=t0
cv(Hi,v − a2i,v,1)
∣∣∣∣∣∣
κ/2
= O(N).
Hence Lemma 7.2 follows from Lyapunov’s theorem(cf. Petrov (1995, p. 122)
).
�
Lemma 7.3. If H0 and Assumptions 2.1–2.7 hold, then we have for all t0 ≤ t ≤ T − t0that
N∑
i=1
S⊤i,tCi,t,1Ri,t = oP (N
1/2),N∑
i=1
S⊤i,tCi,t,3Ri,T = oP (N
1/2)
andN∑
i=1
S⊤i,TCi,t,2Ri,T = oP (N
1/2),
N∑
i=1
R⊤i,tCi,t,3Si,T = oP (N
1/2),
where t0 and t0 are defined in Assumption 3.1.
Proof. Using Assumptions 2.1 – 2.3 we conclude that E S⊤i,tCi,t,1Ri,t = 0 and
(7.5) var
(N∑
i=1
S⊤i,tCi,t,1Ri,t
)=
N∑
i=1
E (S⊤i,tCi,t,1Ri,t)
2.
Next we note that (S⊤i,tCi,t,1Ri,t)
2 ≤ ‖Si,t‖2‖Ci,t,1‖2‖Ri,t‖2 ≤ c‖Si,t‖2‖Ri,t‖2 with someconstant on account of Assumption 2.6 – 2.7. Using again Assumption 2.6 we get thatE ‖Si,t‖2 ≤ cσ2
i and E ‖Ri,t‖2 ≤ c‖γi‖2 with some constant c. On account of Assump-tion 2.2(ii) we conclude
26
var
(N∑
i=1
S⊤i,tCi,t,1Ri,t
)= O(1)
N∑
i=1
σ2i ‖γi‖2 = O(1)
N∑
i=1
‖γi‖2,
and therefore Chebyshev’s inequality and Assumption 2.4 imply the first statement inLemma 7.3. The same argument can be used to prove the other upper bounds.
�
Lemma 7.4. If H0 and Assumptions 2.1 and 2.3 – 2.7 hold, then we have for all t0 ≤t ≤ T − t0 that
N∑
i=1
R⊤i,tCi,t,1Ri,t = oP (N
1/2),N∑
i=1
R⊤i,TCi,t,2Ri,T = oP (N
1/2)
andN∑
i=1
R⊤i,tCi,t,3Ri,T = oP (N
1/2),
where t0 and t0 are defined in Assumption 3.1.
Proof. Using Assumptions 2.6 and 2.7 we get that∣∣∣∣∣
N∑
i=1
R⊤i,tCi,t,ℓRi,t
∣∣∣∣∣ ≤N∑
i=1
t∑
s=1
t∑
s′=1
∣∣∣∣x⊤i,sCi,t,1xi,s′γ
⊤i Λsγ
⊤i Λs′
∣∣∣∣
= OP (1)
N∑
i=1
‖γi‖2,
and therefore the first part follows from Assumption 2.4. The remaining three cases canbe proven in a similar way and therefore the details are omitted.
�
Proof of Theorem 3.1. The result follows immediately from Lemmas 7.1–7.4. �
To prove Theorem 3.2 we note that
ei,t = wi,t + ri,t, where wi,t = εi,t − x⊤i,tZ
−1i,TSi,T and ri,t = γ⊤
i Λt − x⊤i,tZ
−1i,TJi,T
with
Ji,t =
t∑
v=1
xi,vγ⊤i Λv.
Hence
e2i,t = w2i,t + 2wi,tri,t + r2i,t.
Lemma 7.5. If H0 and Assumptions 2.1 – 2.6 and 3.2 hold, then we have
{N−1/2
N∑
i=1
(t∑
s=1
w2i,s − a2i,t,2
), 1 ≤ t ≤ T
}D→{ξ(2)t , 1 ≤ t ≤ T
},
where ξ(2)t , 1 ≤ t ≤ T is jointly normal with zero mean and covariance Γ(2)(t, t′).
27
Proof. It is easy to see that E wi,s = σ2i (1−x⊤
i,sZ−1i,Txi,s). It follows from Assumptions 2.2,
2.5 and 2.6 that E |w2i,s − E w2
i,s|κ/2 ≤ cE εκi,s with some c and therefore
(7.6)
N∑
i=1
E
∣∣∣∣∣
t∑
s=1
w2i,s − a2i,t,2
∣∣∣∣∣
κ/2
= O(N).
As in the proof of Lemma 7.2, we can assume without loss of generality that Γ(2)(t, r′) isnon–singular. Hence for all (c1, c2, . . . , cT )
⊤ 6= 0 we have that
limN→∞
1
N
N∑
i=1
var
(T∑
t=1
ct
(t∑
s=1
w2i,s − a2i,t,2
))> 0.
We get from (7.6) that
N∑
i=1
E
∣∣∣∣∣
T∑
t=1
ct
(t∑
s=1
w2i,s − a2i,t,2
)∣∣∣∣∣
κ/2
= O(N).
Now the lemma follows from Lyapunov’s theorem (cf. Petrov (1995, p. 122)) and theCramer–Wold lemma (cf. Billingsley (1968)). �
Lemma 7.6. If H0 and Assumptions 2.1–2.6 and 3.2 hold, then we have for all 1 ≤ t ≤ Tthat
N∑
i=1
r2i,t = oP (N1/2)
andN∑
i=1
ri,twi,t = oP (N1/2).
Proof. We note that
r2i,t ≤ 2‖γi‖2‖Λt‖2 + 2(x⊤i,tZ
−1i,TJi,T
)2
and by Assumption 2.4N∑
i=1
‖γi‖2‖Λt‖2 = oP (N1/2).
Also, by Assumptions 2.5 and 2.6 we have(x⊤i,tZ
−1i,TJi,T
)2 ≤ c‖Ji,T‖2 ≤ c
and thereforeN∑
i=1
(x⊤i,tZ
−1i,TJi,T
)2= O(1) max
1≤s≤T‖Λs‖2
N∑
i=1
‖γi‖2 = oP (N1/2),
completing the proof of the first part of Lemma 7.6.We write that
wi,tri,t = εi,tγ⊤i Λt − εi,tx
⊤i,t Z
−1i,TJi,T − x⊤
i,tZ−1i,TSi,Tγ
⊤i Λt + x⊤
i,tZ−1i,TSi,Tx
⊤i,tZ
−1i,TJi,T .
Using Assumptions 2.1–2.3 we get that E εi,tγ⊤i,tΛt = 0 and
var
(N∑
i=1
εi,tγ⊤i Λt
)= O(1)
N∑
i=1
‖γi‖2 = O(N1/2)
28
on account of Assumption 2.4. Similarly, E εi,tx⊤i,tZ
−1i,TJi,T = 0 and
var
(N∑
i=1
εi,tx⊤i,tZ
−1i,TJi,T
)= O(N1/2).
Repeating the arguments above, one can verify that∣∣∣∣∣
N∑
i=1
xi,tZ−1i,TSi,Tγ
⊤i Λt
∣∣∣∣∣ = OP (N1/4)
and ∣∣∣∣∣
N∑
i=1
xi,tZ−1i,TSi,Txi,tZ
−1i,TJi,T
∣∣∣∣∣ = OP (N1/4),
completing the proof of the second part of Lemma 7.6.�
Proof of Theorem 3.2. It is an immediate consequence of Lemmas 7.5 and 7.6. �
Proof of Theorem 3.3. It follows from the proof of Lemma 7.2 that
N∑
i=1
Hi,t −A(1)N (t) = OP (N
1/2),
without assuming the existence of the limit in Assumption 3.1, since under the conditionsof Theorem 3.3
lim supN→∞
1
N
∣∣∣∣∣
N∑
i=1
[E{(Z−1
i,t Si,t − Z−1i,TSi,T )
⊤ bCi,t(Z−1i,t Si,t − Z−1
i,TSi,T )
×(Z−1i,t′Si,t′ − Z−1
i,TSi,T )⊤Ci,t′(Z
−1i,t′Si,t′ − Z−1
i,TSi,T )}− a2i,t,1a
2i,t′,1
]∣∣∣∣∣< ∞.
Elementary arguments give
H i,t =t∑
s,v=1
Λ⊤s γix
⊤i,s(Z
−1i,t − Z−1
i,T )Ci,t(Z−1i,t − Z−1
i,T )γ⊤i Λv,
where
H i,t = R⊤i,tCi,t,1Ri,t +R⊤
i,TCi,t,2Ri,T − 2R⊤i,tCi,t,3Ri,T
= γTi
( t∑
j=1
ΛjxTi,jZ
−1i,t −
T∑
j=1
ΛjxTi,jZ
−1i,T
)Ci,t
(Z−1
i,t
t∑
v=1
xivΛTv −Z−1
i,T
T∑
v=1
xivΛTv
)γi
=
t∑
j=1
t∑
v=1
ΛTj γ ix
Ti,j
(Z−1
i,t I{1 ≤ j ≤ t} −Z−1i,T
)C i,t
(Z−1
i,t I{1 ≤ v ≤ t} −Z−1i,T
)xivγ
Ti Λv
Hence the definition of the function Q yields
1
rN
N∑
i=1
H i,t →t∑
s,v=1
Λ⊤s Q(s, v, t)Λv a.s..
29
We showed in the proof of Lemma 7.3 that∣∣∣∣∣
N∑
i=1
S⊤i,tCi,t,1Ri,t
∣∣∣∣∣ = OP (1)
N∑
i=1
‖γi‖2,∣∣∣∣∣
N∑
i=1
S⊤i,TCi,t,2Ri,T
∣∣∣∣∣ = OP (1)
N∑
i=1
‖γi‖2
and ∣∣∣∣∣
N∑
i=1
S⊤i,tCi,t,3Ri,T
∣∣∣∣∣ = OP (1)N∑
i=1
‖γi‖2,∣∣∣∣∣
N∑
i=1
R⊤i,tCi,t,3Si,T
∣∣∣∣∣ = OP (1)N∑
i=1
‖γi‖2.
The proof of Theorem 3.3 is now complete. �
Proof of Theorem 3.4. Elementary arguments give
r2i,t = Λ⊤t γiγ
⊤i Λt − 2
T∑
v=1
Λ⊤v γix
⊤i,vZ
−1i,Txi,tγ
⊤i Λt +
T∑
s,v=1
Λ⊤s γix
⊤i,sZ
−1i,t xi,tx
⊤i,tZ
−1i,t xi,vγ
⊤i Λv.
Using Assumption 3.5 we obtain that{
1
rN
N∑
i=1
r2i,t, 1 ≤ t ≤ T
}→{ξ(4)t , 1 ≤ t ≤ T
}a.s..
It follows from the proof of Theorem 3.2 that
N∑
i=1
w2i,t − A
(2)N (t) = OP (N
1/2).
Minor modification of the proof of Theorem 3.3 gives
N∑
i=1
wi,tri,t = oP (rN),
and therefore (3.15) is proven. �
Proof of Theorem 4.1 Let βi,t and βi,T denote the OLS estimators when γi = 0, 1 ≤i ≤ N and define
Φi,t = (βi,t − βi,T )⊤Cit(βi,t − βi,T )−
1
N
N∑
j=1
(βj,t − βj,T )⊤Cjt(βj,t − βj,T ).
Using Assumption 2.4 one can show that
(7.7)∣∣∣U∗
N(t)−N−1/2N∑
i=1
ζiΦi,t
∣∣∣ = oP (1), t0 ≤ t ≤ t0.
Note that {ζiΦi,t, t0 ≤ t ≤ t0} are independent vectors with zero mean. Let λt0, . . . , λT−t0
be arbitrary constant and define linear combinations
zi = ζi
T−t0∑
t=t0
λtΦi,t.
It follows from Assumption 3.1 along the lines of the proof of Theorem 3.1 that
E( 1√
N
N∑
i=1
zi
)2→
T−t0∑
t=t0
T−t0∑
s=t0
λtΓ(1)(t, s)λs
30
Also, on account of Assumption 2.2 we have that E|zi|1+κ/2 is uniformly bounded.Hence by the Cramer-Wold device (cf. Billingsley, 1968)
{N−1/2
N∑
i=1
ζiΦi,t, t0 ≤ t ≤ t0
}D→{ξ(1)t , t0 ≤ t ≤ t0
}.
To prove the second part of the theorem we note that (7.7) holds under alternatives.It is easy to see that
